Local maximal operators on fractional Sobolev spaces

Abstract

Abstract. In this note we establish the boundedness properties of local maximal operators M G on the fractional Sobolev spaces W s,p (G) whenever G is an open set in R n , 0 < s < 1 and 1 < p < ∞. As an application, we characterize the fractional (s, p)-Hardy inequality on a bounded open set G by a Maz'ya-type testing condition localized to Whitney cubes.

Beyond Local Maximal Operators

“…In particular, Theorem 1.1 generalizes a recently obtained boundedness result for the local Hardy-Littlewood maximal operator M dist(·,∂G) on fractional Sobolev spaces W s,p (G), see [37,Theorem 1.1]. Another interesting case is when R is an α-Hölder function (0 < α < 1) on a bounded open set G such that 0 ≤ R(x) ≤ dist(x, ∂G) for each x ∈ G. Corollary 5.6 then implies that…”

“…Here delicate modifications are required in the proofs due to the A p weight and the R-function. In the sequel, we follow outline of the proof in [37]; in particular, we repeat many details therein without further notice.…”

“…This non-locality contributes to our heavy notation, involving several parameters. However, the number of parameters reduces when we look at special cases: In the light of Example 3.2, the 'relative (s, p)-capacity' These fractional (s, p)-capacities have recently found applications, e.g., in connection with the fractional Hardy inequalities, we refer to [7,37].…”

Hardy-Sobolev inequalities and weighted capacities in metric spaces

“…The proofs in [18] were based on potential theoretic tools such as Harnack inequalities and the existence of capacitary potentials. In this paper, we use a different approach which combines and develops ideas from [9,12,14,20] and is better suited to the weighted (β = 0) case and to exponents q ≥ p. An important feature in the proof of Theorem 5.2 is the use of the so-called discrete convolution as a capacity test function. The definition of the discrete convolution is recalled in Section 4, where we also show that if u belongs to the Newtonian space N 1,p 0 (Ω), then a local maximal function of g u is a p-weak upper gradient of the discrete convolution u t .…”

“…In addition to the results in [18,Theorem 1] and [14,Theorem 10.52], the quasiadditivity property has been considered in the Euclidean space R n for instance in [2,3] for Riesz capacities, and in [9,20] (q = p) and [12] (q ≥ p) for fractional capacities with the help of fractional Hardy(-Sobolev) inequalities. Sufficient conditions for different versions of (weighted) Hardy and Hardy-Sobolev inequalities, and hence also for the corresponding quasiadditivity, have been given for example in [8,14,17,19,21]; see also the references therein.…”